Stuart Findlay’s late header secured 2-1 home victory for Kilmarnock against Celtic in their Scottish Premiership match while Rangers swept St Johnstone aside 5-1 at Ibrox.

Leigh Griffiths gave Celtic a 1-0 first-half lead after Kirk Broadfoot failed to clear Kieran Tierney’s cross after 34 minutes. Greg Taylor hit the post before the Scotland striker found the net with a close-range diving header. Dedryck Boyata came close to adding a second soon afterwards but Jamie MacDonald parried his effort over the bar.

Chris Burke equalised for Killie just after the hour with his long-range strike going in off the post. The away side continued to push for a second goal and MacDonald impressed with a one-handed save to keep Lewis Morgan’s effort out.

But Celtic were sent home pointless after Killie pounced in the third minute of added time. The former Celtic youth player headed Burke’s corner into the bottom corner to hand Kilmarnock maximum points.

Afterwards Brendan Rodgers admitted he is facing the biggest test of his Celtic tenure in what is their worst start to a Premiership season in 20 years. After six games the champions have only 10 points and are six points behind the leaders, Hearts, their poorest start since the 1998-99 campaign under Jozef Venglos.

“It’s a challenge,” he said. “We have to accept that. There’s no doubt we need to be better. When you’re at the biggest clubs and you don’t win, the heat comes on to you. That’s when you show you’re a Celtic player, manager, member of staff. You come together. When you lose games, the spotlight will always be on you. You have to work hard, do the basics right. The basics cost us this time.”

Killie have now gone four games without defeat by Celtic and their manager, Steve Clarke, praised Findlay, the former Parkhead youth player, for playing through the pain barrier. “He didn’t train all week but he put his body on the line,” said Clarke. “He was struggling at half-time but those are some of the rewards the footballing Gods give you.

“Stuart had a bang on the knee and it was sore through the week. He got another bang in the first half but we’ve got good character in the squad. We spoke at half-time about believing. We’ve shown we can compete with the big teams. Second half we competed well and deserved to get back in.”

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At Ibrox Rangers gave another demonstration of their growing confidence as they dispatched St Johnstone 5-1.

Steven Gerrard had to urge his players to believe in themselves as they claimed an impressive Europa League draw with Villarreal on Thursday. But there was not a hint of doubt in their play on Sunday as James Tavernier, Alfredo Morelos, Scott Arfield, Kyle Lafferty and Daniel Candeias fired the goals which propel Rangers to second place in the Premiership.

Blair Alston tucked home a late penalty but it was scant consolation on a painful day for Saints, who remain in eighth place.

Gerrard’s men opened the scoring in the ninth minute. Joe Shaughnessy’s trip on Tavernier handed Rangers a free-kick on the edge of the box and the Ibrox captain made the Irishman pay as he whipped the ball past the wall to leave Zander Clark no chance.

The goalkeeper did much better moments later when Morelos got a lucky bounce as he went past Richard Foster before slipping in Ryan Kent, whose low drive was pushed away well.

Foster then kicked a Lassana Coulibaly header off the line before the former Ibrox defender earned himself a booking for a crude hack on Kent. From the set piece Tavernier came close to repeating his earlier trick as he crashed the ball against the crossbar.

But the second did arrive 11 minutes before the interval. Morelos picked up possession wide on the right and after exchanging passes with Tavernier looked to be heading down a blind alley as Scott Tanser guided him towards the byline. But with a shake of the hips the Colombian fooled Saints’ left-back into lunging at fresh air as he cut back on to his left to smash past Clark.

It was game over on 52 minutes as Rangers wrapped up victory. Morelos sparked a rapid attack with a flick to escape Shaughnessy’s attentions. Kent met his cut-back but his powerful strike rattled the bar. Arfield latched on to the rebound in a flash and kept his shot low as he rifled home the third.

Jon Flanagan was lucky his slack back-pass was not punished by Tony Watt as the former Celtic striker clipped wide of Allan McGregor’s posts.

Clark did well to keep the score down as he turned away efforts from Kent and Arfield but there was no stopping Lafferty’s goal as he climbed off the bench to sweep home from a flowing counterattack on 74 minutes.

Alston earned his side a spot-kick three minutes later as he was tripped by Connor Goldson. He made no mistake from the spot but the deficit was soon back at four as Candeias grabbed number five in the 79th minute, taking his time to finish off after being teed up by Arfield.

“Today in patches was some of the best football I’ve seen from us,” Gerrard told Rangers TV. “We’re trying to do it the Rangers way, which is good, exciting, attacking football.”

“We’re very happy. We asked the players to come in with the right attitude, forget about Thursday night and get the job done. I thought at times we played some really stunning football.

“We certainly deserved to win by a big margin. That’s what I want and that’s what they did today. I’d be interested to know what their [St Johnstone] manager said and what their players think of today.

“That’s what it has to be about here. We have to show people what we’re about. We have to play our way, we have to make it uncomfortable for teams and that looked uncomfortable for St Johnstone, so that’s how I want it.”